How do children learn about infinity?

This is a post in response to a research article titled: INFINITY OF NUMBERS: HOW STUDENTS UNDERSTAND IT:

"Consider the sequence of natural numbers 1, 2, 3, … and think of continuing it on
and on. There is no limit to the process of counting; it has no endpoint."

We know this to be true and that a child could in theory never run out of numbers that they could possibly make up. But because we've already been through our formative and grade school schooling we know that this is true and we have a name for it. Its called infinity and we accept it to be true. On dictionary.com the definitions of infinity are:

infinity

in·fin·i·ty

1 .the quality or state of being infinite.

2.something that is infinite.

3.infinite space, time, or quantity.

4.an infinite extent, amount, or number.

5.an indefinitely great amount or number.

We know and accept this but do children? Can they understand this? This is an incredibly hard concept for children to wrap their heads around but at the same time a very intriguing one. Many students may struggle with this concept. The author of this study notes that "previous research has identified typical problems and constructive
teaching approaches to cardinality of infinite sets." With that being said how do we teach the concept of infinity?

 

That question may not lie in the application of lesson plans or in the way it is taught. It may lie more introducing this key mathematical concept at a stage in the child's cognitive development when they can truly grasp and grapple with it.  We know from research that as child age and gain more life experience that they increase their capabilities to understand mathematical concepts. The research in this report backs up that claim of development psychologists. In the article data reveals the best time to introduce the concept of infinity to be when they are older and they also found a gender difference stating "Boys give better answers than girls in tasks dealing with infinity."

 

So reflect on when you first were dealing with the concept of infinity. Who told you it was true? When did you first believe it to be true? Did you ever test it? And if so how high of a number did you go before you believed it to be true?

 

Feel free to explore this interesting article for yourself: 

http://www.emis.de/proceedings/PME30/4/345

 

And if you would like further proof of the existence of infinity truly check out this article on infinity:

http://plus.maths.org/content/does-infinity-exist

 

The base that tells time...

When we talk about numbers we understand how our number system is setup. We have negative numbers, we have zero, and we have as many positive numbers as we can conjure up. What we work in everyday is the Arabic number system or what is known as base 10. When we looked at bases we were all comfortable working within our base system.  We can do operations easily and estimate answers because we know this:

0,1,2,3,4,5,6,7,8,9

These are our whole numbers and they make up the majority of the numbers that we deal with on a daily basis. When we get to 9 everyone know that we go to 10. But the simplistic nature of these shouldn’t be overlooked. Essentially this is just grouping by tens.  When we have a group of ten we have “filled” up a place value therefore we need to create a new category. We use this concept every day and every time we interact with numbers; and for the most part we don’t appreciate the usefulness of this concept.

So what would happen if we weren’t in base ten, well I’m glad you asked. How about if well look at a base that was invented by the ancient Sumerians in the 3rd millennium BC, base 60. And what if I told you that you, yes you, interact with it every day, would you believe me?

We should first define what base 60 is. Base 60 is commonly referred to as the sexagesimal system. The ancient Babylonians invented this trivial. They choose the number 60 because of this highly composite nature. Remember a composite number is a number that can be divided by a number other than 1 and itself. 60 can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. When we talked above about how when we have nine of something we regrouped into another group similarly in base 60 when we have a grouping of 59 things we have but if we have 60 things we have one grouping of 60 and zero groups of one and we recognize this number: 10. Now we have to add the sub notion which helps us distinguish between our base 10 and base 60. So if we have 60 things in base 60 that number would be: 10_base60.

So your probably asking well then what if we had 72 items then the number should be broken down by saying we have 1 grouping of 60 and we have 12 things left so the number should be 140_base60 right? Wrong. Instead when we go above 9 we have to use symbols to represent numbers, the easiest way to do this is to use the alphabet. So once we go above 9, A=10, B=11, C=12 … and so on. So the number 72 in base 60 should be written like so: 1C_base60. 

Now I mentioned before that we actually use this in our everyday lives. But when? Lets think about when we group things in terms of 60 things…. We do it all the TIME! When we tell time we use the sexagesimal system; 60 secs is a 1 minute, 60 mins is 1 hour. We also use it in relation to circles in a specific type of math, trigonometry. So the next time you overlook the simplistic nature of base 10 and say its hard, just remember that we also work in base 60 and could you imagine not only writing very large number but also incorporating numbers as well to represent a number? Talk about confusing!

Play and Math in the Same Sentence?

I recently came across this article: http://www.scholastic.com/teachers/lesson-plan/math-play-how-young-children-approach-math titled, Math Play: How Young Children Approach Math. The article explores how teachers can use play as a way to introduce math concepts to young children.

Some of the strategies that the article mentioned were  classifying, exploring magnitude, enumeration, investigating dynamics, studying patterns and shapes, exploring spatial relations, block building, math through water play, dramatic mathematics, and math through manipulatives.

The article shows that there are many ways to implement math concepts in a way that in non intrusive and that can be presented as a fun and exciting way to engage with them. Children, especially young children, can benefit greatly from this type of introduction to mathematics. I believe that a lot of the apprehension that children associate with math is that it is not fun. But it doesn't need to be presented through boring worksheets and tests.

The article gives a brief insight in to how teachers can promote math in everyday play. Some of which we have looked as a developmentally appropriate way to do so. Observing children play and intervening sensitively to discuss and clarify ideas. Also a key point the author spends some time on is scheduling enough time for block play and the benefits of block play. I have include a PDF file the outlines the development of children's mathematical skills in an age wise approach:

http://www.scholastic.com/teachers/lesson-plan/collateral_resources/pdf/ECTonline/MathandPlay-AgebyAge_01-02-05.pdf.pdf

If we can introduce math as a fun and exciting thing rather than an monotonous task that requires worksheets and repetitiveness. This may also combat the latter math anxiety that many people deal with. Its about changing the culture and the vocabulary and the way we introduce math to our young children. Math can be fun, it should be fun!

With the implementation of the common core I feel we are moving away from the model of fun and more towards a model of boring repetitiveness. This is how and why children hate math and get turned off from math. So I asked myself how we can change the culture, how can we make math fun? And I came across this informative video:

http://www.voanews.com/content/teacher-uses-music-to-make-math-fun-cool/1594457.html

This video is gear toward older students but it shows a teacher who uses music, raps, to help students remember math concepts. The use of music can help students remember concepts it keeps math fun and it is a change to the humdrum that math commonly is associated with. Techniques like this should be commonplace in the younger classrooms because children love sing-song, rhyming, and songs in general.

Teachers like the one in the video go above and beyond to make math fun, but if it became a part of math, to make it fun, in the younger grades the little things that we can do to just have a positive association with math can go a long way.

Confused People at Verizon

After listening to the recording on Youtube of a customer trying to explain his rationale of the difference between $0.002 dollars and 0.002 cents it became quickly apparent that there was a breakdown between the customer and the customer service representative. I had to listen to the recording several times because it was a bit confusing because of the two trains of thought. When the video reached the half way point and he was able to talk to the manager he was how to use bigger values to show the difference between dollars and cents. "1 dollar is different from 1 cent," "Half a dollar is different between half a cent." It was clear that the customer was getting frustrated. The customer understand that there was two different units in question, the customer service people didn't seem to grasp that concept. They were correct in their assumption that on paper the two values look similar. But they are indeed different values.

When we look back Chapter 7 and what we covered in class on decimals we looked at place values: tens, hundreds, thousands, etc. And we also looked as fractions and percents. Since we know that we can work seamlessly between decimals, fractions and percents we saw that when we had a decimal, except Pi, it could also be represented as a fraction. And a fraction was a part out of a whole. In this case the man was disputing the fact that 1 dollar is not the same as 1 cent, clearly, its just dollars and sense. One dollar equals 100 pennies so 1 dollar is 100 out of 100 pennies whereas 1 cent is 1 out of 100. The man was trying to example that $0.002 dollars is not in fact the same as 0.002 cents. The second value is far less than the first. The two values were out of different values. Like the customer stated, there was no conversion.

It is important to teach the concept of decimals to elementary school children so they don't end up like this customer service rep at Verizon. But also on a serious note in this case it would have probably been more helpful to explain that $0.002 dollars is out of one whole dollars and 0.002 cents in out of one whole cent. In that way it is clear that the values are not equal. I don't necessarily believe that the decimal was the confusion I think it was more so the units that were involved. It didn't help that the although they were dealing with money the units were not the same. They should have converted the .002 cents to dollars or vice versa so that they were the same units. Similar to how proportions are setup.

Math and Technology

Recently in the Family Life section of the Poughkeepsie Journal there was an article titled "Math apps that measure up" written by Jinny Gudmundsen. The article was written on Oct. 1, 2013.

After reading the article I began to think at how much easier my life is because of technology and how much easier math is because of technology. But that there raises the question... is easier necessarily a better thing?

I came up with a list of just a few technological advances that make math easier:
Mathematica, Online tutoring, whiteboard, graphing calculator, wolfram alpha. There are many many more and in many different niches. All of the these programs work off of complex algorithms all designed to compute mathematical equations quickly, neatly, and easily.

Since we live in an age of social media and technology this should be expected that there are a products that make math easier but is that a good thing? What is your opinion? Does technology hamper or help?

Personally I think that sometimes working with technology and allowing it to solve the problem takes away from the person actually understanding the math behind why they got the answer they got or why this theorem works. Technology is great but it shouldn't be the main method. We should be able to know what we should have to leave for a tip at a restaurant or what 20% off at our favorite department store is without having to take out our cellphones and type it in.

VOTE or COMMENT, there is a poll on the left side!

Mathophobia, Reality or Laziness?

It seems as if whenever there is an unsolvable problem whether it be in math class, on a homework assignment many people choose to give up and declare, "I can't do this," or "I'm not good at math." From this anxiety, or lack of motivation, about math we get this thing called mathophobia. Math as a phobia? Are people legitimately scared of math? I think not. 
Rather I believe people are afraid of the outcomes that are associated with math, specifically the reaction they'll receive from their peers. Many people simply don't try in fear of being chastised because of a wrong answer. So the safer thing to do, rather than be wrong, is to not do the problem that way there is no way to receive any negative feedback. It is sad and I've seen it far too much as I'm sure there are others who have seen it. Its also sad how math has be gendered, meaning that many females feel it is something they simply "can't" do as if they are somehow genetically inept, you're not, you're very capable. 

I argue this point because people can't really be scared of math. We use math everyday whether we realize it or not. We may not be doing crazy calculations that consist of letters complex, invisible or indivisible numbers. But everyday we interact with numbers, at the gas pump, making change for purposes, shopping for clothes, obeying the speed limit, setting the thermostat, changing the channel, checking our bank accounts, making phone calls etc. Our entire day consumed with numbers. So why is it that when it comes to a math class with structured class times and real math problems that follow a # people seem to shutdown. It would be really ridiculous is someone said they had readophobia wouldn't it?

The culprit is laziness. For a good portion of my school years when it came to math class I just did the problems to get them done, not really taking in the process by which I came to the final answer. I assume many other students did the same. And I feel pretty confident with I have my fancy handy dandy trusty TI-84 calculator with me. But take it away and what happens? Suddenly we are stripped of our math skills, as if the possession of the calculator so how affects the ability to "do" math. But there in lies the problem we "do" math, what does that even mean? We don't "do" walking, or "do" reading, or "do" video games, or "do"sports. In any other activity we partake we build a base of knowledge and are always constantly learning newer, better ways to do something. But not for math.

No for math many just see a problem deem it as too hard and quit. The fear of failure is more powerful than so called fear of math. I said earlier that we are surrounded by numbers and we interact with them everyday, every minute. The way we interact with numbers has changed. When the bill comes after eating out at a nice restaurant, how many of you take your iPhone out or your cellphone, swipe to the calculator to figure out what a 20% tip would be on a $78.34 check? Technology while it has greatly improved almost every aspect of our lives has also desensitised our thinking and understand. We have to get back to the basics, we have to understand why 2+2=4, we have to understand the importance of zero, and what a fraction is. If we are going to be successful teachers and if we want our students to be successful we have to reverse this trend.

Mathophobia isn't a real thing, its only as real as we make it. There's no reason to be afraid of math, we shouldn't be afraid fail because that is the best way to learn. Once we realize this and get past the common pitfalls that are associated with math everyone will be better off.